
// Make newform 800.4.c.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_800_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_800_c_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_800_4_c_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_800_4_c_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_800_c();"
function MakeCharacter_800_c()
    N := 800;
    order := 2;
    char_gens := [351, 101, 577];
    v := [2, 2, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_800_c_Hecke();"
function MakeCharacter_800_c_Hecke(Kf)
    N := 800;
    order := 2;
    char_gens := [351, 101, 577];
    char_values := [[1, 0], [1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 4;
    raw_aps := [[0, 0], [0, 1], [0, 0], [0, 3], [60, 0], [0, -25], [0, -15], [-40, 0], [0, -89], [-166, 0], [20, 0], [0, 5], [-250, 0], [0, -71], [0, 107], [0, -245], [800, 0], [250, 0], [0, -387], [100, 0], [0, 115], [1320, 0], [0, -491], [-874, 0], [0, -155], [-1498, 0], [0, -701], [0, 597], [-650, 0], [0, 755], [0, 623], [2660, 0], [0, 1385], [560, 0], [2350, 0], [580, 0], [0, -655], [0, -931], [0, 363], [0, -1625], [1120, 0], [-2842, 0], [3180, 0], [0, 2335], [0, -1495], [4240, 0], [-4060, 0], [0, 2811], [0, 777], [-1134, 0], [0, 855], [-4440, 0], [-850, 0], [-660, 0], [0, -3795], [0, -381], [150, 0], [-6580, 0], [0, 2265], [6950, 0], [0, 1941], [0, -685], [0, 2053], [2220, 0], [0, 4715], [0, -3235], [900, 0], [0, 265], [0, -207], [-8614, 0], [0, 1135], [-8080, 0], [0, 1187], [0, -905], [-8120, 0], [0, 5891], [4350, 0], [0, -3735], [11698, 0], [3650, 0], [1120, 0], [4850, 0], [-12580, 0], [0, -6565], [-8560, 0], [0, 2129], [-2550, 0], [0, -3355], [-14482, 0], [0, -81], [0, -7987], [10760, 0], [0, -4633], [2860, 0], [-7160, 0], [0, 699], [-7446, 0], [-16438, 0], [0, 3661], [10878, 0], [0, 8057], [0, 1845], [0, 1281], [6050, 0], [8260, 0], [0, -8435], [0, -483], [0, -13145], [11640, 0], [-25450, 0], [0, 8347], [0, -7945], [0, -615], [10840, 0], [-14060, 0], [-17650, 0], [0, -13679], [0, -3393], [0, 4515], [15600, 0], [16850, 0], [0, 3995], [0, 9345], [0, -9591], [-23380, 0], [11850, 0], [-25646, 0], [30280, 0], [0, 8723], [0, 8375], [36560, 0], [0, 15071], [11860, 0], [0, 18505], [-11718, 0], [-4706, 0], [0, 14335], [0, 10217], [0, 1965], [4854, 0], [-13140, 0], [22050, 0], [0, -7289], [0, -18527], [6150, 0], [8200, 0], [0, 21495], [0, 16065], [-15440, 0], [0, -23469], [0, -15615], [25550, 0], [0, -2159], [0, 883], [0, 20953], [25140, 0], [-32920, 0], [-10150, 0], [0, 14265], [9678, 0], [0, 18493], [0, 1675], [0, 21887], [8740, 0], [0, -24155], [0, -1141], [-31580, 0], [0, -1395]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_800_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_800_4_c_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_800_4_c_f(:prec:=2)
    chi := MakeCharacter_800_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_800_4_c_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_800_4_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_800_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![4, 0, 1]>,<11,R![-60, 1]>],Snew);
    return Vf;
end function;